A296608 a(n) = BarnesG(3*n).

0, 1, 288, 125411328000, 6658606584104736522240000000, 792786697595796795607377086400871488552960000000000000

Offset: 0

Author

Vaclav Kotesovec, Dec 16 2017

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Barnes G-Function.
• Wikipedia, Barnes G-function

Crossrefs

Cf. A000178, A268504, A296607, A306651.

Programs

Mathematica

Table[BarnesG[3*n], {n, 0, 10}]
Round[Table[Glaisher^8 * E^(-2/3) * 3^(9*n^2/2 - 3*n + 5/12) * (2*Pi)^(1 - 3*n) * BarnesG[n] * BarnesG[n + 1/3]^2 * BarnesG[n + 2/3]^3 * BarnesG[n + 1]^2 * BarnesG[n + 4/3], {n, 0, 10}]]

Formula

a(n) = A^8 * exp(-2/3) * 3^(9*n^2/2 - 3*n + 5/12) * (2*Pi)^(1 - 3*n) * BarnesG(n) * BarnesG(n + 1/3)^2 * BarnesG(n + 2/3)^3 * BarnesG(n+1)^2 * BarnesG(n + 4/3), where A is the Glaisher-Kinkelin constant A074962.

a(n) ~ 3^(9*n^2/2 - 3*n + 5/12) * n^(9*n^2/2 - 3*n + 5/12) * (2*Pi)^((3*n-1)/2) / (A * exp(27*n^2/4 - 3*n - 1/12)), where A is the Glaisher-Kinkelin constant A074962.

a(n) = A000178(3*n-2). - R. J. Mathar, Jul 24 2025